# CHAPTER 14 NONPARAMETRIC TESTS

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 CHAPTER 14 NONPARAMETRIC TESTS Everything that we have done up until now in statistics has relied heavily on one major fact: that our data is normally distributed. We have been able to make inferences about population means (one-sample, two-sample z and t tests and analysis of variance), but in each case we assumed that our population was normal. What happens when we want to perform a test on our data, but we have no idea what its true distribution is, and therefore can t assume that our data are normally distributed? In this case, we use what are called nonparametric tests. These tests do not require any specific form for the distribution of the population. Example #1 Let s say that Stephen wanted to go hiking through a wooded area and had no idea what types of trees or conditions he might find. So he decided to bring with him a multipurpose pocket knife, which had a can opener, scissors, bottle opener and a few cutting blades. Stephen didn t know what to expect and in case of an emergency, or if he got trapped, he d be able to open some of the cans of food and bottles that he had in his bag, and he d also be able to cut through some material, albeit very slowly. But what if Stephen knew that there would be huge branches that he d need to cut away in order to cross through the woods? In this case, he d be much better off with a knife that was more specific to his cutting needs, say an axe. However, if he got trapped by some chance, he wouldn t be able to open his tins of food as well, nor his bottles. So this example is pretty similar to the idea behind nonparametric tests. When we don t know what type of conditions we are facing (what type of distribution we are dealing with) it is more beneficial to bring with a multipurpose knife (or to use a nonspecific distribution nonparametric test). However, although this knife might let you do more things, it doesn t perform as well as a knife made for specific uses. On the other hand, if Stephen were to use an axe in unknown conditions, he wouldn t be able to use it for many things, and if he were to use the axe to say open his bottle, chances are it would break. However, if Stephen knew that he would only need a knife for a specific condition, say to cut away branches, then his axe would come in very handy and would do the job well. So if we know what distribution we are dealing with, it is much more practical and useful to use a particular test that is designed for your specific purpose and conditions. If we know we are dealing with a normally distributed population, it is more beneficial to use a Z, t, or F test when performing an inference about means, and our results will be more accurate.

2 When we don t know what population we are dealing with, it is more beneficial to use a test that can work for any type of distribution, because that way you can be sure that you will be prepared for any condition. At the same time, though, if your population really was normally distributed and you used a nonparametric test, then your results won t be as accurate had you used a Z, t or F test. When our data is normally distributed, the mean is equal to the median and we use the mean as our measure of center. However, if our data is skewed, then the median is a much better measure of center. Therefore, just like the Z, t and F tests made inferences about the population mean(s), nonparametric tests make inferences about the population median(s). We are going to be focusing on nonparametric tests which are rank tests. In these types of tests, we rank (or place in order) each observation from our data set. Although we aren t specifying which distribution our data is from, it must be from a continuous distribution. That is, each distribution must be described by a density curve that allows observations to take on any value in some interval. The following is a table which identifies a particular normal test and its nonparametric (or rank) counterpart.

3 WILCOXON RANK SUM TEST The Wilcoxon Rank Sum test is used to test for a difference between two samples. It is the nonparametric counterpart to the two-sample Z or t test. Instead of comparing two population means, we compare two population medians. Draw an SRS of size n 1 from population 1, and then draw an independent SRS of size n 2 from population 2. So the total number of observations is N = n 1 + n 2. The next step in this test is to rank our set of observations. Although we are dealing with two samples, when we rank the observations, we rank them as if they came from one large group. When N is equal to our total sample size, our smallest observation receives a rank of 1, and the largest observation receives a rank of N. Working with ranks instead of numerical outcomes, allows us to abandon specific assumptions about the shape of the distribution. The sum of the ranks of the first sample is W, the Wilcoxon Rank Sum test statistic. If one sample is truly bigger than the other, we d expect its ranks to be higher than the others. So after we have ranked all of the observations, we sum up the ranks for each of the two samples and we can then compare the two rank sums. If there is no difference between our two samples and our sample sizes are equal, then we d expect W to be roughly half of N (N/2). If our sample sizes are different, then we d expect W to be n1( N + 1) roughly equal to its mean/expected value ( µ w = ). When there truly is a 2 difference between the two samples, then W would be a value far from its mean. If both of our samples come from the same continuous distribution, then W has: n1( N + 1) - Mean µ w = 2 nn 1 2( N+ 1) - Standard Deviation σ w = 12 We generally use words, rather than symbols to express the null and alternative hypotheses. As usual, our null hypothesis is that there is no difference between our two populations, and our alternative hypothesis specifies how we think the two populations are different (whether one-sided or two-sided). P-values are calculated by software, or using a normal approximation. We can form another Z statistic by standardizing W. Then once we have our Z statistic, we can find our p-value using the same methods as Stat I.

4 W is standardized by: Z Example #2 n1( N + 1) W µ W w = = 2 σ w nn 1 2( N+ 1) 12 Many states are considering lowering the blood-alcohol level at which a driver is designated as driving under the influence (DUI) of alcohol. An investigator for a legislative committee designed the following test to study the effect of alcohol on reaction time. Ten participants consumed a specified amount of alcohol. Another group of ten participants consumed the same amount of a nonalcoholic drink, a placebo. The two groups did not know whether they were receiving alcohol or the placebo. The twenty participants average reaction times (in seconds) to a series of simulated driving situations are reported in the following table. A boxplot of the two samples show that the population distributions are skewed right. Does it appear that alcohol consumption increases reaction time? Perform a significance test and clearly state your hypotheses, test statistic, p-value and conclusion. Placebo Alcohol

5 TIES When we find ties in our group of observations (two or more observations have the same value) how do we decide which gets the higher rank? We don t, and instead we assign the same rank to all of our ties, so that each one contributes the same amount to the sum of the ranks for its group. Therefore, we assign the average of the all of the ranks that the ties occupy to them. For example, Observation: Rank: So in this case, had the two 158s been different, they would have received ranks of 3 and 4. Since they are equal, they each are assigned a rank of the average of 3 and 4 (3.5) and the remaining numbers continue being ranked as if 3 and 4 were assigned. Example #3 Rank this set of observations. Sample 1 Sample

6 WILCOXON SIGNED RANK TEST The Wilcoxon Signed Rank test is the nonparametric equivalent to the one-sample Z or t test and the matched pairs test. It is used when we want to make inferences about the mean of one population or the mean difference between two populations in a matched pairs setting. Draw an SRS of size n from a population for a matched pairs study and for each pair find the difference between the two responses. Then rank the absolute value of the differences. Then group all of the positive differences and the negative differences separately. The sum of the ranks of the positive differences is W+, the Wilcoxon Signed Rank test statistic. nn ( + 1) nn ( + 1)(2n+ 1) W+ has mean µ W + = and standard deviation σ W + = If there is a difference between our matched pairs, then we d expect W+ to be far from its mean (or the expected value). To find the p-value, we need to standardize W+, which is nn ( + 1) W+ done by finding Z, where Z = 4. Once we have Z, we can find the nn ( + 1)(2n+ 1) 24 appropriate p-value. When ties are present among the pairs, we assign average ranks. However, ties that occur within pairs, giving a difference equal to 0 for that pair, don t add anything to our test statistic and are therefore dropped from our sample. Example #4 Eight subjects were asked to perform a simple puzzle assembly under normal conditions and under conditions of stress. During the stress condition the subjects were told that a mild shock would be delivered 3 minutes after the start of the experiment and every 30 seconds thereafter until the task was completed. Blood pressure readings were taken under both conditions. Data in the accompanying table represent the highest reading during the experiment. Do the data present sufficient evidence to indicate higher blood pressure readings during conditions of stress? Perform a significance test and clearly state your hypotheses, test statistic, p-value and conclusion. Subject Normal Stress

7 KRUSKAL-WALLIS TEST When we can assume that our data is normally distributed and that the population standard deviations are equal, we can test for a difference among several populations by using the One-way ANOVA F test. However, when our data is not normal, or we aren t sure if it is, we can use the nonparametric Kruskal-Wallis test to compare more than two populations as long as our data come from a continuous distribution. In the One-way ANOVA F test, we are testing to see if our population means are equal. Since our data might not necessarily be symmetric in the nonparametric setting, it is better to use the median as the measure of center, and so in the Kruskal-Wallis test we are testing to see if our population medians are equal. Recall the analysis of variance idea: we write the total observed variation in the responses as the sum of two parts, one measuring variation among the groups (sum of squares for groups, SSG) and one measuring variation among individual observations within the same group (sum of squares for error, SSE). The ANOVA F test rejects the null hypothesis that the mean responses are equal in all groups if SSG is large relative to SSE. The idea of the Kruskal-Wallis rank test is to rank all the responses from all groups together and then apply one-way ANOVA to the ranks rather than to the original observations. If there are N observations in all, the ranks are always the whole numbers from 1 to N. The total sum of squares for the ranks is therefore a fixed number no matter what the data are. So we do not need to look at both SSG and SSE. Although it isn t obvious without some unpleasant algebra, the Kruskal-Wallis test statistic is essentially just SSG for the ranks. When SSG is large, that is evidence that the groups differ. Draw independent SRSs of sizes n 1,n 2,...,n I from I populations. There are N observations in all. Rank all N observations and let R i be the sum of the ranks for the i th sample. The 2 12 Ri Kruskal-Wallis statistic is H = 3( N 1) N( N + 1) n + i When the sample sizes n i are large and all I populations have the same continuous distribution, H has approximately the chi-square distribution with I-1 degrees of freedom. The Kruskal-Wallis test rejects the null hypothesis that all populations have the same distribution when H is large. So like the Wilcoxon rank sum statistic, the Kruskal-Wallis test statistic is based on the sums of the ranks for the groups we are comparing. The more different these sums are, the stronger is the evidence that responses are systematically larger in some groups than in others. As usual, we again assign average ranks to tied observations.

8 Example #5 A psychologist is trying to determine if there is a difference in three methods of training six-year-old children to learn a foreign language. A random selection of 10 six-year-old children with similar backgrounds is assigned to each of three different methods. Method 1 uses the traditional teaching format. Method 2 uses repeated listening to tapes of the language along with classroom instruction. Method 3 uses videotapes exclusively. At the end of a 6-week period, the children were given identical, standardized exams. The exams were scored with high scores indicating a better grasp of the language. Because of drop outs, method 1 had 7 students finishing, method 2 had 8, and method 3 only 6. it is, however, important to note that we must assume that children dropped out for reasons unrelated to performance. The data are given in the following table. Please conduct a significance test to determine if there is a difference between the three teaching methods, when we assume our data are not normally distributed. Teaching Method 1 - traditional 2 tapes + classroom 3 - tapes n 1 =7 n 2 =8 n 3 =6

### Supplement on the Kruskal-Wallis test. So what do you do if you don t meet the assumptions of an ANOVA?

Supplement on the Kruskal-Wallis test So what do you do if you don t meet the assumptions of an ANOVA? {There are other ways of dealing with things like unequal variances and non-normal data, but we won

### 1.5 Oneway Analysis of Variance

Statistics: Rosie Cornish. 200. 1.5 Oneway Analysis of Variance 1 Introduction Oneway analysis of variance (ANOVA) is used to compare several means. This method is often used in scientific or medical experiments

### Nonparametric Statistics

1 14.1 Using the Binomial Table Nonparametric Statistics In this chapter, we will survey several methods of inference from Nonparametric Statistics. These methods will introduce us to several new tables

### LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING

LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING In this lab you will explore the concept of a confidence interval and hypothesis testing through a simulation problem in engineering setting.

### Chapter 16 Appendix. Nonparametric Tests with Excel, JMP, Minitab, SPSS, CrunchIt!, R, and TI-83-/84 Calculators

The Wilcoxon Rank Sum Test Chapter 16 Appendix Nonparametric Tests with Excel, JMP, Minitab, SPSS, CrunchIt!, R, and TI-83-/84 Calculators These nonparametric tests make no assumption about Normality.

### 3. Nonparametric methods

3. Nonparametric methods If the probability distributions of the statistical variables are unknown or are not as required (e.g. normality assumption violated), then we may still apply nonparametric tests

### Inferential Statistics

Inferential Statistics Sampling and the normal distribution Z-scores Confidence levels and intervals Hypothesis testing Commonly used statistical methods Inferential Statistics Descriptive statistics are

### Non-parametric tests I

Non-parametric tests I Objectives Mann-Whitney Wilcoxon Signed Rank Relation of Parametric to Non-parametric tests 1 the problem Our testing procedures thus far have relied on assumptions of independence,

### Nonparametric tests, Bootstrapping

Nonparametric tests, Bootstrapping http://www.isrec.isb-sib.ch/~darlene/embnet/ Hypothesis testing review 2 competing theories regarding a population parameter: NULL hypothesis H ( straw man ) ALTERNATIVEhypothesis

### Analysis of numerical data S4

Basic medical statistics for clinical and experimental research Analysis of numerical data S4 Katarzyna Jóźwiak k.jozwiak@nki.nl 3rd November 2015 1/42 Hypothesis tests: numerical and ordinal data 1 group:

### 1 Nonparametric Statistics

1 Nonparametric Statistics When finding confidence intervals or conducting tests so far, we always described the population with a model, which includes a set of parameters. Then we could make decisions

### QUANTITATIVE METHODS BIOLOGY FINAL HONOUR SCHOOL NON-PARAMETRIC TESTS

QUANTITATIVE METHODS BIOLOGY FINAL HONOUR SCHOOL NON-PARAMETRIC TESTS This booklet contains lecture notes for the nonparametric work in the QM course. This booklet may be online at http://users.ox.ac.uk/~grafen/qmnotes/index.html.

### Good luck! BUSINESS STATISTICS FINAL EXAM INSTRUCTIONS. Name:

Glo bal Leadership M BA BUSINESS STATISTICS FINAL EXAM Name: INSTRUCTIONS 1. Do not open this exam until instructed to do so. 2. Be sure to fill in your name before starting the exam. 3. You have two hours

### Hypothesis testing S2

Basic medical statistics for clinical and experimental research Hypothesis testing S2 Katarzyna Jóźwiak k.jozwiak@nki.nl 2nd November 2015 1/43 Introduction Point estimation: use a sample statistic to

### Recall this chart that showed how most of our course would be organized:

Chapter 4 One-Way ANOVA Recall this chart that showed how most of our course would be organized: Explanatory Variable(s) Response Variable Methods Categorical Categorical Contingency Tables Categorical

### Outline of Topics. Statistical Methods I. Types of Data. Descriptive Statistics

Statistical Methods I Tamekia L. Jones, Ph.D. (tjones@cog.ufl.edu) Research Assistant Professor Children s Oncology Group Statistics & Data Center Department of Biostatistics Colleges of Medicine and Public

### Non-Parametric Tests (I)

Lecture 5: Non-Parametric Tests (I) KimHuat LIM lim@stats.ox.ac.uk http://www.stats.ox.ac.uk/~lim/teaching.html Slide 1 5.1 Outline (i) Overview of Distribution-Free Tests (ii) Median Test for Two Independent

### Comparing Means in Two Populations

Comparing Means in Two Populations Overview The previous section discussed hypothesis testing when sampling from a single population (either a single mean or two means from the same population). Now we

### Nonparametric tests these test hypotheses that are not statements about population parameters (e.g.,

CHAPTER 13 Nonparametric and Distribution-Free Statistics Nonparametric tests these test hypotheses that are not statements about population parameters (e.g., 2 tests for goodness of fit and independence).

### Nonparametric Test Procedures

Nonparametric Test Procedures 1 Introduction to Nonparametrics Nonparametric tests do not require that samples come from populations with normal distributions or any other specific distribution. Hence

### THE KRUSKAL WALLLIS TEST

THE KRUSKAL WALLLIS TEST TEODORA H. MEHOTCHEVA Wednesday, 23 rd April 08 THE KRUSKAL-WALLIS TEST: The non-parametric alternative to ANOVA: testing for difference between several independent groups 2 NON

### Study Guide for the Final Exam

Study Guide for the Final Exam When studying, remember that the computational portion of the exam will only involve new material (covered after the second midterm), that material from Exam 1 will make

### Stat 411/511 THE RANDOMIZATION TEST. Charlotte Wickham. stat511.cwick.co.nz. Oct 16 2015

Stat 411/511 THE RANDOMIZATION TEST Oct 16 2015 Charlotte Wickham stat511.cwick.co.nz Today Review randomization model Conduct randomization test What about CIs? Using a t-distribution as an approximation

### How to Conduct a Hypothesis Test

How to Conduct a Hypothesis Test The idea of hypothesis testing is relatively straightforward. In various studies we observe certain events. We must ask, is the event due to chance alone, or is there some

### Rank-Based Non-Parametric Tests

Rank-Based Non-Parametric Tests Reminder: Student Instructional Rating Surveys You have until May 8 th to fill out the student instructional rating surveys at https://sakai.rutgers.edu/portal/site/sirs

### STAT 350 Practice Final Exam Solution (Spring 2015)

PART 1: Multiple Choice Questions: 1) A study was conducted to compare five different training programs for improving endurance. Forty subjects were randomly divided into five groups of eight subjects

### Power & Effect Size power Effect Size

Power & Effect Size Until recently, researchers were primarily concerned with controlling Type I errors (i.e. finding a difference when one does not truly exist). Although it is important to make sure

### Null Hypothesis H 0. The null hypothesis (denoted by H 0

Hypothesis test In statistics, a hypothesis is a claim or statement about a property of a population. A hypothesis test (or test of significance) is a standard procedure for testing a claim about a property

### Unit 24 Hypothesis Tests about Means

Unit 24 Hypothesis Tests about Means Objectives: To recognize the difference between a paired t test and a two-sample t test To perform a paired t test To perform a two-sample t test A measure of the amount

### PRESIDENTIAL SURNAMES: A NONRANDOM SEQUENCE?

CHAPTER 14 Nonparametric Methods PRESIDENTIAL SURNAMES: A NONRANDOM SEQUENCE? Girl Ray/Getty Images In a series of numbers, names, or other data, is it possible that the series might exhibit some nonrandom

### Unit 29 Chi-Square Goodness-of-Fit Test

Unit 29 Chi-Square Goodness-of-Fit Test Objectives: To perform the chi-square hypothesis test concerning proportions corresponding to more than two categories of a qualitative variable To perform the Bonferroni

### General Method: Difference of Means. 3. Calculate df: either Welch-Satterthwaite formula or simpler df = min(n 1, n 2 ) 1.

General Method: Difference of Means 1. Calculate x 1, x 2, SE 1, SE 2. 2. Combined SE = SE1 2 + SE2 2. ASSUMES INDEPENDENT SAMPLES. 3. Calculate df: either Welch-Satterthwaite formula or simpler df = min(n

### Null and Alternative Hypotheses. Lecture # 3. Steps in Conducting a Hypothesis Test (Cont d) Steps in Conducting a Hypothesis Test

Lecture # 3 Significance Testing Is there a significant difference between a measured and a standard amount (that can not be accounted for by random error alone)? aka Hypothesis testing- H 0 (null hypothesis)

### 496 STATISTICAL ANALYSIS OF CAUSE AND EFFECT

496 STATISTICAL ANALYSIS OF CAUSE AND EFFECT * Use a non-parametric technique. There are statistical methods, called non-parametric methods, that don t make any assumptions about the underlying distribution

### Testing Group Differences using T-tests, ANOVA, and Nonparametric Measures

Testing Group Differences using T-tests, ANOVA, and Nonparametric Measures Jamie DeCoster Department of Psychology University of Alabama 348 Gordon Palmer Hall Box 870348 Tuscaloosa, AL 35487-0348 Phone:

### Data Analysis. Lecture Empirical Model Building and Methods (Empirische Modellbildung und Methoden) SS Analysis of Experiments - Introduction

Data Analysis Lecture Empirical Model Building and Methods (Empirische Modellbildung und Methoden) Prof. Dr. Dr. h.c. Dieter Rombach Dr. Andreas Jedlitschka SS 2014 Analysis of Experiments - Introduction

### 7.1 Inference for comparing means of two populations

Objectives 7.1 Inference for comparing means of two populations Matched pair t confidence interval Matched pair t hypothesis test http://onlinestatbook.com/2/tests_of_means/correlated.html Overview of

### Research Methods 1 Handouts, Graham Hole,COGS - version 1.0, September 2000: Page 1:

Research Methods 1 Handouts, Graham Hole,COGS - version 1.0, September 000: Page 1: NON-PARAMETRIC TESTS: What are non-parametric tests? Statistical tests fall into two kinds: parametric tests assume that

### Variables and Data A variable contains data about anything we measure. For example; age or gender of the participants or their score on a test.

The Analysis of Research Data The design of any project will determine what sort of statistical tests you should perform on your data and how successful the data analysis will be. For example if you decide

### Inferential Statistics. Probability. From Samples to Populations. Katie Rommel-Esham Education 504

Inferential Statistics Katie Rommel-Esham Education 504 Probability Probability is the scientific way of stating the degree of confidence we have in predicting something Tossing coins and rolling dice

### Statistical Inference and t-tests

1 Statistical Inference and t-tests Objectives Evaluate the difference between a sample mean and a target value using a one-sample t-test. Evaluate the difference between a sample mean and a target value

### Two-sample hypothesis testing, II 9.07 3/16/2004

Two-sample hypothesis testing, II 9.07 3/16/004 Small sample tests for the difference between two independent means For two-sample tests of the difference in mean, things get a little confusing, here,

### Testing: is my coin fair?

Testing: is my coin fair? Formally: we want to make some inference about P(head) Try it: toss coin several times (say 7 times) Assume that it is fair ( P(head)= ), and see if this assumption is compatible

### Statistics. One-two sided test, Parametric and non-parametric test statistics: one group, two groups, and more than two groups samples

Statistics One-two sided test, Parametric and non-parametric test statistics: one group, two groups, and more than two groups samples February 3, 00 Jobayer Hossain, Ph.D. & Tim Bunnell, Ph.D. Nemours

### AP STATISTICS 2009 SCORING GUIDELINES (Form B)

AP STATISTICS 2009 SCORING GUIDELINES (Form B) Question 5 Intent of Question The primary goals of this question were to assess students ability to (1) state the appropriate hypotheses, (2) identify and

### Independent samples t-test. Dr. Tom Pierce Radford University

Independent samples t-test Dr. Tom Pierce Radford University The logic behind drawing causal conclusions from experiments The sampling distribution of the difference between means The standard error of

### 1 Confidence intervals

Math 143 Inference for Means 1 Statistical inference is inferring information about the distribution of a population from information about a sample. We re generally talking about one of two things: 1.

### c. The factor is the type of TV program that was watched. The treatment is the embedded commercials in the TV programs.

STAT E-150 - Statistical Methods Assignment 9 Solutions Exercises 12.8, 12.13, 12.75 For each test: Include appropriate graphs to see that the conditions are met. Use Tukey's Honestly Significant Difference

### Section 9.3B Inference for Means: Paired Data

Section 9.3B Inference for Means: Paired Data 1 Objective PERFORM significance tests for paired data are called: paired t procedures. Comparative studies (i.e. 2 observations on 1 individual or 1 observation

### THE FIRST SET OF EXAMPLES USE SUMMARY DATA... EXAMPLE 7.2, PAGE 227 DESCRIBES A PROBLEM AND A HYPOTHESIS TEST IS PERFORMED IN EXAMPLE 7.

THERE ARE TWO WAYS TO DO HYPOTHESIS TESTING WITH STATCRUNCH: WITH SUMMARY DATA (AS IN EXAMPLE 7.17, PAGE 236, IN ROSNER); WITH THE ORIGINAL DATA (AS IN EXAMPLE 8.5, PAGE 301 IN ROSNER THAT USES DATA FROM

### Chapter 3: Nonparametric Tests

B. Weaver (15-Feb-00) Nonparametric Tests... 1 Chapter 3: Nonparametric Tests 3.1 Introduction Nonparametric, or distribution free tests are so-called because the assumptions underlying their use are fewer

### Unit 26 Estimation with Confidence Intervals

Unit 26 Estimation with Confidence Intervals Objectives: To see how confidence intervals are used to estimate a population proportion, a population mean, a difference in population proportions, or a difference

### Name: Date: Use the following to answer questions 3-4:

Name: Date: 1. Determine whether each of the following statements is true or false. A) The margin of error for a 95% confidence interval for the mean increases as the sample size increases. B) The margin

### NONPARAMETRIC STATISTICS 1. depend on assumptions about the underlying distribution of the data (or on the Central Limit Theorem)

NONPARAMETRIC STATISTICS 1 PREVIOUSLY parametric statistics in estimation and hypothesis testing... construction of confidence intervals computing of p-values classical significance testing depend on assumptions

### AP Statistics 1998 Scoring Guidelines

AP Statistics 1998 Scoring Guidelines These materials are intended for non-commercial use by AP teachers for course and exam preparation; permission for any other use must be sought from the Advanced Placement

### SCHOOL OF HEALTH AND HUMAN SCIENCES DON T FORGET TO RECODE YOUR MISSING VALUES

SCHOOL OF HEALTH AND HUMAN SCIENCES Using SPSS Topics addressed today: 1. Differences between groups 2. Graphing Use the s4data.sav file for the first part of this session. DON T FORGET TO RECODE YOUR

### For example, enter the following data in three COLUMNS in a new View window.

Statistics with Statview - 18 Paired t-test A paired t-test compares two groups of measurements when the data in the two groups are in some way paired between the groups (e.g., before and after on the

### Difference tests (2): nonparametric

NST 1B Experimental Psychology Statistics practical 3 Difference tests (): nonparametric Rudolf Cardinal & Mike Aitken 10 / 11 February 005; Department of Experimental Psychology University of Cambridge

### Chapter 7 Part 2. Hypothesis testing Power

Chapter 7 Part 2 Hypothesis testing Power November 6, 2008 All of the normal curves in this handout are sampling distributions Goal: To understand the process of hypothesis testing and the relationship

### Lesson 1: Comparison of Population Means Part c: Comparison of Two- Means

Lesson : Comparison of Population Means Part c: Comparison of Two- Means Welcome to lesson c. This third lesson of lesson will discuss hypothesis testing for two independent means. Steps in Hypothesis

### Class 19: Two Way Tables, Conditional Distributions, Chi-Square (Text: Sections 2.5; 9.1)

Spring 204 Class 9: Two Way Tables, Conditional Distributions, Chi-Square (Text: Sections 2.5; 9.) Big Picture: More than Two Samples In Chapter 7: We looked at quantitative variables and compared the

### Descriptive Statistics

Descriptive Statistics Primer Descriptive statistics Central tendency Variation Relative position Relationships Calculating descriptive statistics Descriptive Statistics Purpose to describe or summarize

### Simple Regression Theory II 2010 Samuel L. Baker

SIMPLE REGRESSION THEORY II 1 Simple Regression Theory II 2010 Samuel L. Baker Assessing how good the regression equation is likely to be Assignment 1A gets into drawing inferences about how close the

### Tutorial 5: Hypothesis Testing

Tutorial 5: Hypothesis Testing Rob Nicholls nicholls@mrc-lmb.cam.ac.uk MRC LMB Statistics Course 2014 Contents 1 Introduction................................ 1 2 Testing distributional assumptions....................

### Post-hoc comparisons & two-way analysis of variance. Two-way ANOVA, II. Post-hoc testing for main effects. Post-hoc testing 9.

Two-way ANOVA, II Post-hoc comparisons & two-way analysis of variance 9.7 4/9/4 Post-hoc testing As before, you can perform post-hoc tests whenever there s a significant F But don t bother if it s a main

### Business Statistics. Lecture 8: More Hypothesis Testing

Business Statistics Lecture 8: More Hypothesis Testing 1 Goals for this Lecture Review of t-tests Additional hypothesis tests Two-sample tests Paired tests 2 The Basic Idea of Hypothesis Testing Start

### Hypothesis Testing Level I Quantitative Methods. IFT Notes for the CFA exam

Hypothesis Testing 2014 Level I Quantitative Methods IFT Notes for the CFA exam Contents 1. Introduction... 3 2. Hypothesis Testing... 3 3. Hypothesis Tests Concerning the Mean... 10 4. Hypothesis Tests

### BA 275 Review Problems - Week 6 (10/30/06-11/3/06) CD Lessons: 53, 54, 55, 56 Textbook: pp. 394-398, 404-408, 410-420

BA 275 Review Problems - Week 6 (10/30/06-11/3/06) CD Lessons: 53, 54, 55, 56 Textbook: pp. 394-398, 404-408, 410-420 1. Which of the following will increase the value of the power in a statistical test

### II. DISTRIBUTIONS distribution normal distribution. standard scores

Appendix D Basic Measurement And Statistics The following information was developed by Steven Rothke, PhD, Department of Psychology, Rehabilitation Institute of Chicago (RIC) and expanded by Mary F. Schmidt,

### Measures of Central Tendency and Variability: Summarizing your Data for Others

Measures of Central Tendency and Variability: Summarizing your Data for Others 1 I. Measures of Central Tendency: -Allow us to summarize an entire data set with a single value (the midpoint). 1. Mode :

### Introduction. Chapter 14: Nonparametric Tests

2 Chapter 14: Nonparametric Tests Introduction robustness outliers transforming data other standard distributions nonparametric methods rank tests The most commonly used methods for inference about the

### ACTM State Exam-Statistics

ACTM State Exam-Statistics For the 25 multiple-choice questions, make your answer choice and record it on the answer sheet provided. Once you have completed that section of the test, proceed to the tie-breaker

### Two-sample t-tests. - Independent samples - Pooled standard devation - The equal variance assumption

Two-sample t-tests. - Independent samples - Pooled standard devation - The equal variance assumption Last time, we used the mean of one sample to test against the hypothesis that the true mean was a particular

### Nonparametric Two-Sample Tests. Nonparametric Tests. Sign Test

Nonparametric Two-Sample Tests Sign test Mann-Whitney U-test (a.k.a. Wilcoxon two-sample test) Kolmogorov-Smirnov Test Wilcoxon Signed-Rank Test Tukey-Duckworth Test 1 Nonparametric Tests Recall, nonparametric

### Hypothesis Testing: Two Means, Paired Data, Two Proportions

Chapter 10 Hypothesis Testing: Two Means, Paired Data, Two Proportions 10.1 Hypothesis Testing: Two Population Means and Two Population Proportions 1 10.1.1 Student Learning Objectives By the end of this

### The Wilcoxon Rank-Sum Test

1 The Wilcoxon Rank-Sum Test The Wilcoxon rank-sum test is a nonparametric alternative to the twosample t-test which is based solely on the order in which the observations from the two samples fall. We

### Using Kruskal-Wallis to Improve Customer Satisfaction. A White Paper by. Sheldon D. Goldstein, P.E. Managing Partner, The Steele Group

Using Kruskal-Wallis to Improve Customer Satisfaction A White Paper by Sheldon D. Goldstein, P.E. Managing Partner, The Steele Group Using Kruskal-Wallis to Improve Customer Satisfaction KEYWORDS Kruskal-Wallis

### Statistiek I. Nonparametric Tests. John Nerbonne. CLCG, Rijksuniversiteit Groningen.

Statistiek I Nonparametric Tests John Nerbonne CLCG, Rijksuniversiteit Groningen http://www.let.rug.nl/nerbonne/teach/statistiek-i/ John Nerbonne 1/36 Overview 1 Mann-Whitney U-Test 2 Wilcoxon s Signed

### " Y. Notation and Equations for Regression Lecture 11/4. Notation:

Notation: Notation and Equations for Regression Lecture 11/4 m: The number of predictor variables in a regression Xi: One of multiple predictor variables. The subscript i represents any number from 1 through

### HYPOTHESIS TESTING: POWER OF THE TEST

HYPOTHESIS TESTING: POWER OF THE TEST The first 6 steps of the 9-step test of hypothesis are called "the test". These steps are not dependent on the observed data values. When planning a research project,

### Come scegliere un test statistico

Come scegliere un test statistico Estratto dal Capitolo 37 of Intuitive Biostatistics (ISBN 0-19-508607-4) by Harvey Motulsky. Copyright 1995 by Oxfd University Press Inc. (disponibile in Iinternet) Table

### STA218 Introduction to Hypothesis Testing

STA218 Introduction to Hypothesis Testing Al Nosedal. University of Toronto. Fall 2015 October 29, 2015 Who wants to be a millionaire? Let s say that one of you is invited to this popular show. As you

### Hypothesis Testing COMP 245 STATISTICS. Dr N A Heard. 1 Hypothesis Testing 2 1.1 Introduction... 2 1.2 Error Rates and Power of a Test...

Hypothesis Testing COMP 45 STATISTICS Dr N A Heard Contents 1 Hypothesis Testing 1.1 Introduction........................................ 1. Error Rates and Power of a Test.............................

### Statistiek I. t-tests. John Nerbonne. CLCG, Rijksuniversiteit Groningen. John Nerbonne 1/35

Statistiek I t-tests John Nerbonne CLCG, Rijksuniversiteit Groningen http://wwwletrugnl/nerbonne/teach/statistiek-i/ John Nerbonne 1/35 t-tests To test an average or pair of averages when σ is known, we

### UNIVERSITY OF NAIROBI

UNIVERSITY OF NAIROBI MASTERS IN PROJECT PLANNING AND MANAGEMENT NAME: SARU CAROLYNN ELIZABETH REGISTRATION NO: L50/61646/2013 COURSE CODE: LDP 603 COURSE TITLE: RESEARCH METHODS LECTURER: GAKUU CHRISTOPHER

### reductio ad absurdum null hypothesis, alternate hypothesis

Chapter 10 s Using a Single Sample 10.1: Hypotheses & Test Procedures Basics: In statistics, a hypothesis is a statement about a population characteristic. s are based on an reductio ad absurdum form of

### Wilcoxon Rank Sum or Mann-Whitney Test Chapter 7.11

STAT Non-Parametric tests /0/0 Here s a summary of the tests we will look at: Setting Normal test NonParametric Test One sample One-sample t-test Sign Test Wilcoxon signed-rank test Matched pairs Apply

### Two-Sample T-Tests Allowing Unequal Variance (Enter Difference)

Chapter 45 Two-Sample T-Tests Allowing Unequal Variance (Enter Difference) Introduction This procedure provides sample size and power calculations for one- or two-sided two-sample t-tests when no assumption

### 9-3.4 Likelihood ratio test. Neyman-Pearson lemma

9-3.4 Likelihood ratio test Neyman-Pearson lemma 9-1 Hypothesis Testing 9-1.1 Statistical Hypotheses Statistical hypothesis testing and confidence interval estimation of parameters are the fundamental

### 13: Additional ANOVA Topics. Post hoc Comparisons

13: Additional ANOVA Topics Post hoc Comparisons ANOVA Assumptions Assessing Group Variances When Distributional Assumptions are Severely Violated Kruskal-Wallis Test Post hoc Comparisons In the prior

### Comparing two groups (t tests...)

Page 1 of 33 Comparing two groups (t tests...) You've measured a variable in two groups, and the means (and medians) are distinct. Is that due to chance? Or does it tell you the two groups are really different?

### One-Sample t-test. Example 1: Mortgage Process Time. Problem. Data set. Data collection. Tools

One-Sample t-test Example 1: Mortgage Process Time Problem A faster loan processing time produces higher productivity and greater customer satisfaction. A financial services institution wants to establish

### Section: 101 (10am-11am) 102 (11am-12pm) 103 (1pm-2pm) 104 (1pm-2pm)

Stat 0 Midterm Exam Instructor: Tessa Childers-Day 1 May 014 Please write your name and student ID below, and circle your section. With your signature, you certify that you have not observed poor or dishonest

### Solutions 7. Review, one sample t-test, independent two-sample t-test, binomial distribution, standard errors and one-sample proportions.

Solutions 7 Review, one sample t-test, independent two-sample t-test, binomial distribution, standard errors and one-sample proportions. (1) Here we debunk a popular misconception about confidence intervals

### One-Way Analysis of Variance (ANOVA) Example Problem

One-Way Analysis of Variance (ANOVA) Example Problem Introduction Analysis of Variance (ANOVA) is a hypothesis-testing technique used to test the equality of two or more population (or treatment) means

### HYPOTHESIS TESTING WITH SPSS:

HYPOTHESIS TESTING WITH SPSS: A NON-STATISTICIAN S GUIDE & TUTORIAL by Dr. Jim Mirabella SPSS 14.0 screenshots reprinted with permission from SPSS Inc. Published June 2006 Copyright Dr. Jim Mirabella CHAPTER

### ANOVA MULTIPLE CHOICE QUESTIONS. In the following multiple-choice questions, select the best answer.

ANOVA MULTIPLE CHOICE QUESTIONS In the following multiple-choice questions, select the best answer. 1. Analysis of variance is a statistical method of comparing the of several populations. a. standard

### Statistics: revision

NST 1B Experimental Psychology Statistics practical 5 Statistics: revision Rudolf Cardinal & Mike Aitken 3 / 4 May 2005 Department of Experimental Psychology University of Cambridge Slides at pobox.com/~rudolf/psychology